Combining individually valid and conditionally i.i.d. P-variables

TL;DR

This paper develops a method to combine valid P-variables, especially order statistics, into a single valid P-variable, with applications demonstrated in biological association studies.

Contribution

It introduces the function $f_{n,k}$ for optimally combining order statistic P-variables into a valid summary, extending the theory of P-value combination under specific independence conditions.

Findings

The function $f_{n,k}$ is the smallest increasing function ensuring validity.

An approximation for $f_{n,k}$ is given by $1 \, \wedge \, \frac{n}{k} u$ for large $k$.

Application to primate association data demonstrates practical utility.

Abstract

For a given testing problem, let $U_1,...,U_n$ be individually valid and conditionally on the data i.i.d.\ P-variables (often called P-values). For example, the data could come in groups, and each $U_i$ could be based on subsampling just one datum from each group in order to satisfy an independence assumption under the hypothesis. The problem is then to deterministically combine the $U_i$ into a valid summary P-variable. Restricting here our attention to functions of a given order statistic $U_{k:n}$ of the $U_i$, we compute the function $f_{n,k}$ which is smallest among all increasing functions $f$ such that $f(U_{k:n})$ is always a valid P-variable under the stated assumptions. Since $f_{n,k}(u)\le 1\wedge (\frac {n}{k} u)$, with the right hand side being a good approximation for the left when $k$ is large, one may in particular always take the minimum of 1 and twice the left sample median of the given P-variables. We sketch the original application of the above in a recent study of associations between various primate species by Astaras et al.